# Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

^{*}

^{*}Corresponding author.

- DOI
- 10.1080/14029251.2020.1819608How to use a DOI?
- Keywords
- lattice Schwarzian Korteweg-de Vries equation; integrable symplectic map; finite genus solution
- Abstract
Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the discrete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation are calculated through Riemann surface method.

- Copyright
- © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

Remarkable progress has been made in recent years in the study of discrete soliton equations (see [12] and the references therein). Among the related mathematical theories, the property of multi-dimensional consistency plays an important role in the understanding of discrete integrability. In the 2-dimensional case, it leads to the well-known Adler-Bobenko-Suris (ABS) list [1, 10], which gives a classification of integrable quadrilateral lattice equations. Quite a few works have appeared in the study of the ABS equations, concerning their relations with the usual soliton equations, the Lax pairs, explicit analytic solutions, Bäcklund transformations (BTs), symmetries and conservation laws etc. [3, 6, 7 , 13, 14, 19, 21, 22, 26, 32, 33 ].

The purpose of this paper is to investigate the lSKdV equation, which was first given in [23],

*u*=

*u*(

*m*,

*n*),

*û*=

*u*(

*m*,

*n*+1). Eq. (1.1) is exactly the Q1(0) model (Q1 with

*δ*= 0) in the ABS hierarchy. The approach of Lax representation will be used to confirm the integrability of Eq. (1.1) and to calculate its basic explicit analytic solutions, the finite genus solutions [6, 7].

To produce a purely discrete Lax pair, it is vital to select two appropriate discrete spectral problems. It turns out that a special role is played by the semi-discrete integrable equations, which are also of independent interest, see [18] and references therein, where the well-known Toda, the Volterra and the Ablowitz-Ladik hierarchies are investigated thoroughly. A semi-discrete Lax pair can be constructed with the help of a continuous spectral problem and its Darboux transformation (DT), where the DT is regarded as a discrete spectral problem [4, 18], which usually leads to an integrable symplectic map by using the non-linearization technique [5–7]. Refer to [16], integrable maps are called BTs whose geometrical explanation is given in terms of spectral curves and their Jacobians. And the symplectic correspondences (BTs) compatible with finite gap solutions of KdV have been discussed through DTs for the standard KdV spectral problem [15].

In our case we consider the continuous SKdV equation,

*S*[

*ϕ*;

*x*] denotes the Schwarzian derivative of

*ϕ*[20, 30, 31], i.e.

Technically, it is more convenient to use the derivative version

*w*=

*ϕ*. The Eq. (1.4) has a Lax pair given by

_{x}We note that in [24], the Lax pair for one Schwarzian PDE, which is equivalent to the SKdV hierarchy via expansions on the independent variables and has a fully discrete counterpart (1.1) by considering the independent variables as lattice parameters, has been found. However, we have not been able to blend it with the algebro-geometric technique of nonlinearization employed in the present paper. Fortunately, here each of the linear systems (1.5) and (1.6) can be nonlinearized to produce an integrable Hamiltonian system. Thus we find a Liouville integrable system associated with a spectral problem (see Sec. 2) given by

The compatibility condition

This suggests a constraint *b* = *γ*/(*u* − *ũ*) and leads to a Lax pair, different from the one in [23], for Eq. (1.1).

### Lemma 1.1.

*The lSKdV equation* (1.1) *has a Lax pair*

*with*

*where*

*and*Ξ

*is defined by*

*Eq.*(1.1).

The paper is organised as follows. In Sec. 2, a finite-dimensional Hamiltonian system which is a nonlinear version of the spectral problem (1.7) is presented. In Sec. 3, resorting to the Hamiltonian system, we construct an integrable symplectic map. In addition, with the help of the Burchnall-Chaundy theory, the discrete potential is expressed in terms of theta functions. In Sec. 4, based on the discrete version of the Liouville-Arnold theorem, the finite genus solutions of lSKdV equation (1.1) are obtained through the commutativity of integrable maps [7].

## 2. The Integrable Hamiltonian System (*H*_{1})

Take the symplectic manifold (ℝ^{2}* ^{N}*, d

*p*∧ d

*q*) as the phase space. The symplectic coordinate is defined as (

*p*,

*q*) = (

*p*

_{1},...,

*p*,

_{N}*q*

_{1},...,

*q*). Let

_{N}*A*= diag(

*α*

_{1},...,

*α*) with distinct, non-zero

_{N}*σ*

_{+},

*σ*

_{3}are the usual Pauli matrices, and

∀ (ξ,η) = (ξ_{1},...,ξ* _{N}*,η

_{1},...,η

*) ∈ ℝ*

_{N}*.*

^{2N}The generating function *ℱ _{λ}* = det

*L*(

*λ*;

*p*,

*q*) is a rational function of the argument

*ζ*=

*λ*

^{2},

The expansion

*l*= 1, 2,...), where

*H*

_{1}), defined by the Hamiltonian function

They are exactly *N* copies of Eq. (1.7) with distinct *λ* = *α _{j}* and the constraint

In this context (*H*_{1}) is called a non-linearization of the linear spectral problem (1.7).

According to the Liouville-Arnold theory [2], we shall discuss the coefficients *F*_{1},...,*F _{N}* given by (2.3) are first integrals of the phase flow with Hamiltonian function

*H*

_{1}, i.e., {

*F*,

_{j}*H*

_{1}} = 0 (

*j*= 1,...,

*N*), where {·, ·} denotes the Poisson bracket on the phase space. The involution and functional independence between

*F*

_{1},...,

*F*guarantee that the Hamiltonian system (

_{N}*H*

_{1}) is completely integrable.

Consider the Hamiltonian system (*ℱ _{λ}*),

*L*(

*λ*) is the abbreviation of

*L*(

*λ*;

*p*,

*q*) and

*L*(

^{ij}*λ*),

*i*,

*j*= 1, 2 are entries of the matrix

*L*(

*λ*). Hence we obtain d

*ε*/d

_{j}*t*= [

_{λ}*W*(

*λ*,

*α*),

_{j}*ε*], where [·, ·] stands for the matrix commutator. Based on this formula, it is easy to derive the following basic equation,

_{j}As a corollary, we have

Actually, by Eq. (2.7), (d/d*t _{λ}*)

*L*

^{2}(

*μ*) = [

*W*(

*λ*,

*μ*),

*L*

^{2}(

*μ*)]. Since

*L*

^{2}(

*μ*) = −

*Iℱ*, where

_{μ}*I*is the identity matrix, we have d

*ℱ*/d

_{μ}*t*= 0. According to the definition of Poisson bracket [2], this is exactly Eq. (2.8), whose power series expansion gives rise to Eq. (2.9).

_{λ}The generating function *ℱ _{λ}* has a factorization

*R*(

*ζ*) =

*ζα*(

*ζ*)

*Z*(

*ζ*), where

*F*

_{1}is given by Eq. (2.3). The spectral curve is defined as

*g*=

*N*− 1 and has two points at infinity, ∞

_{+}, ∞

_{−}. At the branch point 𝔬 = (

*ζ*= 0,

*ξ*= 0),

*ℛ*has a local coordinate

*λ*=

*ζ*

^{1/2}. The generic point on

*ℛ*is given as

*τ*:

*ℛ*→

*ℛ*is the hyperelliptic involution. The variables

Putting *μ* = *ν _{k}*, with

*ℱ*-flow,

_{λ}Eq. (2.15) is rewritten in a simple form and gives rise to

### Proposition 2.1.

*The ℱ _{λ}- and the F_{l}-flow are linearized by ϕ′_{s}*

*as*

*where A*

_{0}= 1;

*A*−

*= 0 (*

_{j}*j*= 1, 2,...); while

*A*(

_{j}*j*= 1, 2,...), are defined by

In particular, {*ϕ′ _{s}*,

*F*1} = 0, 1 ⩽

*s*⩽

*g*.

### Proposition 2.2.

*The Hamiltonian system* (*H*_{1}) *is integrable, possessing N integrals F*_{1},...,*F _{N}, involutive with each other and functionally independent in the dense, open subset 𝒪* = {(

*p*,

*q*) ∈ ℝ

^{2N}:

*F*

_{1}≠ 0}.

### Proof.

*F _{l}* is an integral since {

*H*

_{1},

*F*} = (1/2){

_{l}*F*

_{1},

*F*} = 0 by Eq. (2.9). It needs only to prove that d

_{l}*F*

_{1},...,d

*F*are linearly independent in

_{N}*p*,

*q*) ∈

*𝒪*. Suppose

By Eq. (2.18), the coefficient matrix is non-degenerate,

Thus *c*_{2} = ··· = *c _{N}* = 0 and

*c*

_{1}d

*F*

_{1}= 0. We have

*c*

_{1}= 0 since d

*F*

_{1}≠ 0 at

*𝒪*. Otherwise,

Hence *α _{j}q_{j}* + 〈

*p*,

*q*〉

*p*= 0, ∀

_{j}*; and*

_{j}*F*

_{1}= 0. This is a contradiction.

## 3. The Integrable Symplectic Map *𝒮*_{γ}

_{γ}

As a non-linearization of Eq. (1.8), define a map *𝒮 _{γ}* : ℝ

^{2}

*→ ℝ*

^{N}^{2}

*,*

^{N}*b*=

*f*(

_{γ}*p*,

*q*) is to be chosen so that

*𝒮*is integrable and symplectic.

_{γ}### Lemma 3.1.

*Let P*(* ^{γ}*)(

*b*;

*p*,

*q*) =

*b*

^{2}

*L*

^{21}(

*γ*) + 2

*bL*

^{11}(

*γ*) −

*L*

^{12}(

*γ*)

*. Then*

### Proof.

By Eq. (3.1), we get

Based on these preparations, we calculate the left-hand side of Eq. (3.2),

By using Eq. (3.1), we obtain

This proves Eq. (3.2). Eq. (3.3) is obtained through direct calculations.

Consider the quadratic equation *P*(* ^{γ}*)(

*b*) = 0, whose roots give the constraint on

*b*,

Actually *γb* can be written as a meromorphic function on *ℛ*,

Though doubled-valued as a function of *β* ∈ ℂ, it is single-valued as a function of 𝔭(*β*^{2}) ∈ *ℛ*. Hence we obtain

### Proposition 3.1.

*The map 𝒮 _{γ}* : ℝ

^{2}

*→ ℝ*

^{N}^{2}

*,*

^{N}*, defined as*

*is symplectic and integrable, possessing the Liouville set of integrals*

### Proof.

Since *P*(* ^{γ}*)(

*b*) = 0, by Eq. (3.2) and (3.3) we have

Taking the determinant of Eq. (3.8), we obtain

By Eq. (3.7), the discrete flow *F _{l}*}. Define finite genus potentials as

By Eq. (3.4), they have the relation

*m*-flow, Eq. (3.8) is rewritten as

*L*(

_{m}*λ*) =

*Lλ*;

*p*(

*m*),

*q*(

*m*),

*u*with the help of the following spectral problem and its fundamental solution matrix

_{m}*M*(

_{γ}*m*,

*λ*),

By induction we have

### Lemma 3.2.

*The following functions are polynomials of the argument ζ* = *λ*^{2}*:*

*Besides, as λ* → ∞,

By Eq. (3.13), the solution space *ℰ _{λ}* of Eq. (3.14) is invariant under the action of the linear operator

*L*(

_{m}*λ*), which has two eigenvalues

They define a meromorphic function *ℛ* with

Putting *m* = 0, we solve

*λ*→ ∞,

### Lemma 3.3 (Formula of Dubrovin-Novikov type).

### Proof.

Using Eq. (3.16), we calculate the left-hand side of Eq. (3.25),

With the help of Eq. (2.12) and (3.25), Eq. (3.26) is verified by some calculations.

### Lemma 3.4.

*As λ* → ∞,

### Proof.

Since

Thus we obtain Eq. (3.28) by solving

From Eq. (3.20) we have

By (Lemma 3.2) and the discussion on *H*^{(2)}(2*k*, 𝔭) and *H*^{(2)}(2*k* + 1, 𝔭) are defined on *ℛ*, respectively, with

### Proposition 3.2.

*H*^{(2)}(2*k*, 𝔭) *and H*^{(2)}(2*k* + 1, 𝔭) *have the divisors respectively,*

### Proof.

From Eq. (3.26) and (3.29) we obtain

*ζ*). As 𝔭 → ∞

_{±}, by Eq. (3.28) and (3.29) we have

By these formulas it is easy to calculate the divisors.

By using the technique developed by Toda [28], based on the meromorphic differentials dln*H*^{(2)}(2*k*, 𝔭) and dln*H*^{(2)}(2*k* + 1, 𝔭), immediately we get

*ℛ*, while

*𝒯*is the basic lattice spanned by the periodic vectors of

*ℛ*[8, 11]. With the help of the Abel map

This endows Eq. (3.33) with a clear geometric explanation.

### Proposition 3.3.

*In the Jacobi variety J*(*ℛ*) = ℂ^{g}
/𝒯, the discrete flow*is linearized by the Abel-Jacobi variable*

*where δ*

_{2}

*= 0,*

_{k}*δ*

_{2}

_{k}_{+1}= 1

*, and*

The meromorphic function *H*^{(2)}(2*k*, 𝔭) is expressed by its divisor up to a constant factor

*K*is the Riemann constant vector and

*w*[𝔭, 𝔮] is an Abel differential of the third kind, possessing only two simple poles at 𝔭, 𝔮 with residues +1, −1, respectively. Resorting to Eq. (3.32), by the asymptotic behaviors of Eq. (3.37) near ∞

_{±}we obtain

We introduce a new variable *v _{m}* by

Cancelling the constant factor in Eq. (3.38), we arrive at

Similarly, considering the analytic expression for *H*^{(2)}(2*k* + 1, 𝔭) leads to

### Proposition 3.4.

*The finite genus potential v _{m}, defined by*

*Eq.*(3.11)

*and*(3.40)

*, has an explicit evolution formula along the discrete flow*

*,*

*where the vectors K*(

*m*)

*,*Ω

*Ω*

_{γ},_{0}

*Ω*

_{γ}and*are given by*

*Eq.*(3.36)

*and*(3.42)

*, while the constants R*

_{γ}, R_{0}

_{γ}are defined by*Eq.*(3.42)

*and*(3.44)

*; moreover, δ*

_{2}

*= 0*

_{k}*, δ*

_{2}

_{k}_{+1}= 1

*, for all k.*

## 4. Solutions of lSKdV equation (1.1)

Let *γ*_{1}, *γ*_{2} be the two constants given in Eq. (1.1). By (Proposition 3.1), setting *γ* = *γ*_{1}, *γ*_{2} in the above we have two symplectic maps *𝒮 _{γ}*

_{1}and

*𝒮*

_{γ}_{2}, sharing the same set of integrals {

*F*}. Resorting to the discrete version of Liouville-Arnold theorem [25,27,29], they commute. Thus we have well-defined functions with two discrete arguments

_{l}*m*and

*n*,

### Proof.

By the commutativity of

From Eq. (3.12) we obtain

By Eq. (3.6), *χ _{j}* = (

*p*(

_{j}*m*,

*n*),

*q*(

_{j}*m*,

*n*))

*solves simultaneously*

^{T}Thus *u _{mn}* satisfies Eq. (1.1) by Eq. (1.11). In order to prove that

*v*is also a solution, it is sufficient to notice that (i)

_{mn}*F*

_{1}is a constant of motion which is independent of

*m*and

*n*; (ii) Eq. (1.1) is invariant under the Möbius transformation

*u*↦

*v*given by Eq. (4.1).

Apply Eq. (3.45) to the flow *v*_{00} → *v _{m}*

_{0}→

*v*we obtain

_{mn}### Proposition 4.2.

*The lSKdV equation* (1.1) *has finite genus solutions*

*and*

*. Further, any Möbius transformation wmn*= (

*a*

_{11}

*v*+

_{mn}*a*

_{12})/(

*a*

_{21}

*v*+

_{mn}*a*

_{22})

*solves*

*Eq.*(1.1)

*, where ajk are constants.*

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 11426206; 11501521), State Scholarship Found of China (CSC No. 201907045035), and Graduate Student Education Research Foundation of Zhengzhou University (Grant No. YJSXWKC201913). We would like to thank Prof. Frank W. Nijhoff and Prof. Da-jun Zhang for helpful discussions.

## References

### Cite this article

TY - JOUR AU - Xiaoxue Xu AU - Cewen Cao AU - Guangyao Zhang PY - 2020 DA - 2020/09/04 TI - Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation JO - Journal of Nonlinear Mathematical Physics SP - 633 EP - 646 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819608 DO - 10.1080/14029251.2020.1819608 ID - Xu2020 ER -